PIPE FRICTION One very important effect of fluid friction is experienced in the resistance to the flow of water through a pipe line. This resist ance can only be overcome by a drop in pressure in the direction of motion, and reasoning upon analogy to the resistance to motion over a plane surface it might be inferred that with turbulent motion the total resistance would equal f Sv'1.
Putting A =sectional area of pipe in sq.ft.
A Here A -:-P = area/per imeter is termed the "hydraulic mean radius" and is commonly denoted by "m" so that f'lvn pi-P2 = m For a circular pipe m= 2 irr = d : 4. The earlier experi menters, assuming the loss to be proportional to the kinetic energy, wrote this where C and f are coefficients whose values depend on the ness of the pipe. These formulae, with the appropriate values of f or of C for the particular pipe diameter and velocity are generally used in calculations. More recent investigations in dicate that the coefficient also depends on the pipe diameter and on the velocity of flow, and that the index n is less than 2•0 except in a rough pipe, and tend to show that an exponential formula, Thus two small pipes of diameter d, will give the same discharge as a single large pipe D, of the same length, if Owing to the very convenient form of the Chezy equation, v = C Vmi), it is often an advantage to have at hand values of C cor responding to various diameters and velocities of flow. Such approximate values are given in the following tables : or one pipe 13-1 in. in diameter would give the same discharge as two 1 o in. pipes.
After a period of use the incrustation of a pipe line diminishes its discharge. The rate and type of incrustation depends on the class of water and on the material of the pipe walls. To allow for this diminution, the pipe should be designed to give an initial dis charge in excess of the requirements. The excess percentage dis charge for different types of pipe should be approximately as follows :— Flow Through Pipes Coupled Up in Parallel.—If a series of pipes of diameters etc., discharge in parallel between the same two points, so that the available head h is the same in each case, adopting the relationship.
In general, the coefficient of velocity, of a well-designed nozzle is about •985, and the velocity will be reduced in this ratio.
g g If however the valve closure is sudden, the elasticity of the water is involved. Each layer in turn is brought to rest, its kinetic energy is converted into strain energy, and the disturbance is propagated back to the open end of the pipe with the velocity of sound waves through the me dium, Under these conditions, the phenomenon is known as water hammer, and the rise in pressure p at the valve is obtained from the relationship Here K is the modulus of compressibility of the water, which has a mean value of 43-2 X lb. per square foot. Adopting this value, p= 63.7v lb. per square inch, a value which shows that excessively high pressures may be obtained with comparatively low velocities of flow where this action is set up. In a nonrigid pipe line some energy is expended in stretching the pipe walls, and the hammer pressure is reduced. Taking this into account, K', the effective value of K, is given by This loss may be reduced within limits by tapering the pipe gradually between sections (i) and (2) , and so reducing the velocity gradually. In this case the loss of head is where k has the following mean values.
where r is the radius and t the thickness of the pipe, and for steel pipes E = 43 • 2 X z lb. per square foot and 7==3.6 It may be shown that pressures as great as those corresponding to instantaneous closure are attained if the time of valve closure does not exceed 2l Vp sec. Here the velocity of propagation of pressure waves along the pipe line, is given by V, = and zo is approximately 4,700 ft. per second for a rigid pipe line, but may be as low as 3,000 ft. per second for a large thin-walled pipe line. If the time of closure is greater than 41- -V,, sec., the formula p _ wla lb. per square foot is applicable.
Variation of Velocity over the Cross Section of a Pipe.— The velocity of flow at a cross section of a pipe increases from the walls to the centre. With stream line flow the curve of velocities is a parabola and the velocity at the walls is zero. With turbulent flow the curve of velocities is much flatter near the centre of the pipe. Experiment indicates that even when the flow as a whole is turbulent there is a very thin boundary layer in which the motion is non-turbulent, and that the velocity at the wall itself is zero.
Losses at Valves and Bends in a Pipe Line.—In addition to the losses due to wall friction in a pipe line, there are usually losses due to the presence of valves or bends which upset the normal lines of flow through the pipe. The loss of head due to a partially open valve or sluice is particularly serious, and is largely due to the expansion of the stream section on passing the constric tion. The loss also depends largely on the design, so that values deduced from tests on any one type of valve cannot be taken as applying to another type. The following values have been deter mined experimentally from valves of the types shown in fig. (7 a and b) . Here the loss equals -2g ft., where v is the velocity in the pipe in ft. per sec.
Losses at Bends.—The loss due to a right-angled bend depends - on the radius of curvature R of the bend. The best radius in prac tice is from 2.5 to 50 times the pipe diameter. For such bends the loss is given sufficiently nearly by ft. Where the bend is carried round an angle 9 less than the loss is very nearly pro portional to Loss Due to Enlargement of Section.—When a pipe line has its cross-sectional area suddenly increased from to sq.f t., so that the velocity is reduced from to f .s., violent eddy formation is set up and the con sequent loss of head is given very closely by the expression These losses include the skin friction in the pipe. This accounts for the value of F increasing as the angle of divergence 9 of the sides is diminished below a definite value, about 6° in a circular pipe, and z z ° in a rectangular passage, owing to the increasing length of pipe between points (i) and (2) .
Flow in Pipe Lines—Hydraulic Gradient.—In designing a pipe line, the problem which usually presents itself is that of determining the minimum size of pipe which, with a given loss of head, will discharge a given volume of water per second. The available head is absorbed in giving the kinetic energy of flow in the pipe and overcoming the pipe line losses which are due: 1. to eddy formation at the entrances to the pipe; 2. to bends, valves, changes of sections, etc.; 3. to wall friction.
The loss due to eddy formation at the entrance is small. With a bell mouthpiece it is about ft. With a pipe opening flush with the side of the reservoir it is about ft., and, with a pipe projecting into the reservoir, about ft.
When a submerged open-ended pipe line connects two reser voirs, the velocity of flow in the pipe will adjust itself until the difference of level, h, between their free surfaces is equal to the total head absorbed in these various sources of loss.
If a horizontal be drawn through the upper free water surface and if a series of ordinates be drawn vertically downwards from this to represent on the vertical scale of the drawing the total loss of pressure head from the pipe entrance to the particular point considered, the ends of such ordinates, being joined, give a cprve called the hydraulic gradient for the pipe. If a series of open stand pipes were erected on the pipe line, the free surfaces in these pipes would lie on the gradient line, and the pressure in the pipe is represented, at each point, by its distance below this line. If the pipe is above the gradient line at any point, as is the case in a syphon, the pressure will be less than atmospheric. In order to prevent difficul ties arising from liberation and accumula tion of air at such points, and from ad mission of air at leaky joints, the greatest height above the gradient line should not in any case exceed 20 ft.
Measurement of the Flow in Pipes. —The measurement of the flow in a pipe line may be obtained in several ways. Of these the use of the Venturi meter, the Pitot tube, and in very large pipes the current meter, are the most common. The Pitot tube (fig. 8) consists of a bent tube terminating in a small orifice pointing up- stream, which is surrounded by a second tube whose direction is parallel to that of flow. A series of small holes in the wall of the outer tube admit water, at the mean pressure in their vicinity, to its interior, which is connected to one leg of a manometer. The other leg is connected to the central tube carrying the impact orifice. If v is the velocity of flow im mediately upstream from this orifice, the pressure inside the ori fice, where the velocity is zero, is equal to the sum of the statical pressure at the point, plus ft. of water, where k is a con stant whose value approximates closely to unity in a well-designed tube. It follows that the difference of level of the fluid in the two legs of the manometer equals feet.
For measurements of the flow in pipes the instrument is inserted into the pipe through a gland in the pipe wall. It should be used if possible at a section remote from any bend or source of disturbance. For approximate work the velocity of the central filament may be measured. This when multiplied by a coefficient which varies from •79 in small pipes to .86 in large pipes gives the mean velocity. Alternatively the velocity may be measured at the radius of mean velocity, which varies from about •7a in small pipes to .75a in large pipes, where a is the radius of the pipe. These values, however, only apply to a straight stretch of the pipe, and if it is necessary to make measurements near a bend, and in any case for accurate results the pipe should be traversed along two diameters at right angles, and the velocities measured at a series of radii. If b r is the width of an elementary annulus containing one series of such measurements whose mean value is v, the dis charge is then given by